Making an edge parallel with another
It's difficult to see from your screenshots, but what you may need to do is apply the object rotation and scale before you add that extra vertex.

Try this:


come out of Edit Mode and enter Object Mode again 
with your object selected, hit CTRL+A
now select Rotation and Scale
enter Edit Mode and try again


Without this, Blender will often produce unexpected effects when editing the object further, particularly if you've scaled a primitive in one or two axis before editing it.

Good luck!